The spectrum of global eigenmodes of the Alfven wave omega < omega ci offers the possibility of a very simple method for diagnosing the centre of a Tokamak plasma. The frequencies of these modes depend on the central mass density and the current profile, and can be adequately estimated by MHD theory in cylindrical geometry including ion cyclotron effects. A simple unshielded bar antenna has been used to launch the waves at a power level of a few watts, detected by magnetic field probes in the boundary layer. Applications investigated on the TCA and PETULA Tokamaks include estimates of the core current density over a wide variety of discharges, measurements of changes of the effective mass during impurity puffing and the observation of profile changes during sawtooth activity. A method for the measurement of the deuterium-tritium ratio in a Tokamak is proposed.